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It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy which implies countable choice AND the negation of full choice.

Edit: By "inherited by grothendieck topoi", I mean that for any topos T which satisfies said choice axiom A, the following two properties must hold:

1. For any category C, the category of functors from C to T (presheaves in T on the dual category) must satisfy A.
2. Any topos t which has a geometric embedding in T must also satisfy A (which up to equivalence is the same as restricting T to sheaves with respect to some Lawvere-Tierny topology on T).

Doing 1 and then 2 is basically a generalization of defining Grothendieck topoi over sets but to arbitrary topoi. WISC as I understand it has the property of being preserved by those two operations regardless of which topos you start with.

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy which implies countable choice AND the negation of full choice.

Edit: By "inherited by grothendieck topoi", I mean that for any topos T which satisfies said choice axiom A, the following two properties must hold:

1. For any category C, the category of functors from C to T (which may be viewed as presheaves in T on the dual category of C) must satisfy A.
2. Any topos t which has a geometric embedding in T must also satisfy A (which up to equivalence is the same as restricting T to sheaves with respect to some Lawvere-Tierny topology on T).

Doing 1 and then 2 is basically a generalization of defining Grothendieck topoi over sets but to arbitrary topoi. WISC as I understand it has the property of being preserved by those two operations regardless of which topos you start with.

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy which implies countable choice AND the negation of full choice.

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy which implies countable choice AND the negation of full choice.

Edit: By "inherited by grothendieck topoi", I mean that for any topos T which satisfies said choice axiom A, the following two properties must hold:

1. For any category C, the category of functors from C to T (presheaves in T on the dual category) must satisfy A.
2. Any topos t which has a geometric embedding in T must also satisfy A (which up to equivalence is the same as restricting T to sheaves with respect to some Lawvere-Tierny topology on T).

Doing 1 and then 2 is basically a generalization of defining Grothendieck topoi over sets but to arbitrary topoi. WISC as I understand it has the property of being preserved by those two operations regardless of which topos you start with.

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy.

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy which implies countable choice AND the negation of full choice.